why does the Shroedinger equation have a specific operator for kinetic energy and only a general expression (V) for potential energy

The Schrödinger equation is a fundamental equation in quantum mechanics that describes how the wave function of a quantum system evolves over time. It is a differential equation that relates the energy of the system to the wave function.

The specific operator for kinetic energy in the Schrödinger equation comes from the mathematical concept of the kinetic energy operator in quantum mechanics. In quantum mechanics, physical observables, such as position and momentum, are represented by mathematical operators that act on the wave function. The kinetic energy operator represents the observable of kinetic energy and is derived from the classical expression for kinetic energy by replacing position and momentum with their corresponding quantum mechanical operators.

On the other hand, the potential energy in the Schrödinger equation is represented by a general expression, usually denoted as V. This is because potential energy can take various forms depending on the specific physical system being studied. For example, in the case of a particle in a one-dimensional potential well, the potential energy is usually represented by a piecewise function. In other cases, it might be a continuous function. The general expression for potential energy allows the Schrödinger equation to be applicable to a wide range of physical systems with different potential energy functions.

To further understand the Schrödinger equation and its applications, studying the mathematical formalism of quantum mechanics is necessary. This includes topics such as linear operators, wave functions, and the representation of observables. It is also important to learn about specific potential energy functions to analyze different physical systems in quantum mechanics.